Algebra Examples

$12 x + y^{2} - 14 y + 1 = 0$

Step 1

Rewrite the equation in vertex form.

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Step 1.1

Isolate $x$ to the left side of the equation.

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Step 1.1.1

Move all terms not containing $x$ to the right side of the equation.

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Step 1.1.1.1

Subtract $y^{2}$ from both sides of the equation.

$12 x - 14 y + 1 = - y^{2}$

Step 1.1.1.2

Add $14 y$ to both sides of the equation.

$12 x + 1 = - y^{2} + 14 y$

Step 1.1.1.3

Subtract $1$ from both sides of the equation.

$12 x = - y^{2} + 14 y - 1$

Step 1.1.2

Divide each term in $12 x = - y^{2} + 14 y - 1$ by $12$ and simplify.

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Step 1.1.2.1

Divide each term in $12 x = - y^{2} + 14 y - 1$ by $12$ .

$\frac{12 x}{12} = \frac{- y^{2}}{12} + \frac{14 y}{12} + \frac{- 1}{12}$

Step 1.1.2.2

Simplify the left side.

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Step 1.1.2.2.1

Cancel the common factor of $12$ .

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Step 1.1.2.2.1.1

Cancel the common factor.

$\frac{12 x}{12} = \frac{- y^{2}}{12} + \frac{14 y}{12} + \frac{- 1}{12}$

Step 1.1.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{- y^{2}}{12} + \frac{14 y}{12} + \frac{- 1}{12}$

Step 1.1.2.3

Simplify the right side.

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Step 1.1.2.3.1

Simplify each term.

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Step 1.1.2.3.1.1

Move the negative in front of the fraction.

$x = - \frac{y^{2}}{12} + \frac{14 y}{12} + \frac{- 1}{12}$

Step 1.1.2.3.1.2

Cancel the common factor of $14$ and $12$ .

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Step 1.1.2.3.1.2.1

Factor $2$ out of $14 y$ .

$x = - \frac{y^{2}}{12} + \frac{2 (7 y)}{12} + \frac{- 1}{12}$

Step 1.1.2.3.1.2.2

Cancel the common factors.

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Step 1.1.2.3.1.2.2.1

Factor $2$ out of $12$ .

$x = - \frac{y^{2}}{12} + \frac{2 (7 y)}{2 (6)} + \frac{- 1}{12}$

Step 1.1.2.3.1.2.2.2

Cancel the common factor.

$x = - \frac{y^{2}}{12} + \frac{2 (7 y)}{2 \cdot 6} + \frac{- 1}{12}$

Step 1.1.2.3.1.2.2.3

Rewrite the expression.

$x = - \frac{y^{2}}{12} + \frac{7 y}{6} + \frac{- 1}{12}$

Step 1.1.2.3.1.3

Move the negative in front of the fraction.

$x = - \frac{y^{2}}{12} + \frac{7 y}{6} - \frac{1}{12}$

Step 1.2

Complete the square for $- \frac{y^{2}}{12} + \frac{7 y}{6} - \frac{1}{12}$ .

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Step 1.2.1

Use the form $a x^{2} + b x + c$ , to find the values of $a$ , $b$ , and $c$ .

$a = - \frac{1}{12}$

$b = \frac{7}{6}$

$c = - \frac{1}{12}$

Step 1.2.2

Consider the vertex form of a parabola.

$a {(x + d)}^{2} + e$

Step 1.2.3

Find the value of $d$ using the formula $d = \frac{b}{2 a}$ .

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Step 1.2.3.1

Substitute the values of $a$ and $b$ into the formula $d = \frac{b}{2 a}$ .

$d = \frac{\frac{7}{6}}{2 (- \frac{1}{12})}$

Step 1.2.3.2

Simplify the right side.

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Step 1.2.3.2.1

Multiply the numerator by the reciprocal of the denominator.

$d = \frac{7}{6} \cdot \frac{1}{2 (- \frac{1}{12})}$

Step 1.2.3.2.2

Cancel the common factor of $1$ and $- 1$ .

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Step 1.2.3.2.2.1

Rewrite $1$ as $- 1 (- 1)$ .

$d = \frac{7}{6} \cdot \frac{- 1 (- 1)}{2 (- \frac{1}{12})}$

Step 1.2.3.2.2.2

Move the negative in front of the fraction.

$d = \frac{7}{6} (- \frac{1}{2 (\frac{1}{12})})$

Step 1.2.3.2.3

Combine $2$ and $\frac{1}{12}$ .

$d = \frac{7}{6} (- \frac{1}{\frac{2}{12}})$

Step 1.2.3.2.4

Cancel the common factor of $2$ and $12$ .

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Step 1.2.3.2.4.1

Factor $2$ out of $2$ .

$d = \frac{7}{6} (- \frac{1}{\frac{2 (1)}{12}})$

Step 1.2.3.2.4.2

Cancel the common factors.

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Step 1.2.3.2.4.2.1

Factor $2$ out of $12$ .

$d = \frac{7}{6} (- \frac{1}{\frac{2 \cdot 1}{2 \cdot 6}})$

Step 1.2.3.2.4.2.2

Cancel the common factor.

$d = \frac{7}{6} (- \frac{1}{\frac{2 \cdot 1}{2 \cdot 6}})$

Step 1.2.3.2.4.2.3

Rewrite the expression.

$d = \frac{7}{6} (- \frac{1}{\frac{1}{6}})$

Step 1.2.3.2.5

Multiply the numerator by the reciprocal of the denominator.

$d = \frac{7}{6} (- (1 \cdot 6))$

Step 1.2.3.2.6

Cancel the common factor of $6$ .

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Step 1.2.3.2.6.1

Factor $6$ out of $- (1 \cdot 6)$ .

$d = \frac{7}{6} (6 (- 1 \cdot (1)))$

Step 1.2.3.2.6.2

Cancel the common factor.

$d = \frac{7}{6} (6 (- 1 \cdot (1)))$

Step 1.2.3.2.6.3

Rewrite the expression.

$d = 7 (- 1 \cdot (1))$

Step 1.2.3.2.7

Multiply $- 1$ by $1$ .

$d = 7 \cdot - 1$

Step 1.2.3.2.8

Multiply $7$ by $- 1$ .

$d = - 7$

Step 1.2.4

Find the value of $e$ using the formula $e = c - \frac{b^{2}}{4 a}$ .

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Step 1.2.4.1

Substitute the values of $c$ , $b$ and $a$ into the formula $e = c - \frac{b^{2}}{4 a}$ .

$e = - \frac{1}{12} - \frac{{(\frac{7}{6})}^{2}}{4 (- \frac{1}{12})}$

Step 1.2.4.2

Simplify the right side.

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Step 1.2.4.2.1

Simplify each term.

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Step 1.2.4.2.1.1

Simplify the numerator.

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Step 1.2.4.2.1.1.1

Apply the product rule to $\frac{7}{6}$ .

$e = - \frac{1}{12} - \frac{\frac{7^{2}}{6^{2}}}{4 (- \frac{1}{12})}$

Step 1.2.4.2.1.1.2

Raise $7$ to the power of $2$ .

$e = - \frac{1}{12} - \frac{\frac{49}{6^{2}}}{4 (- \frac{1}{12})}$

Step 1.2.4.2.1.1.3

Raise $6$ to the power of $2$ .

$e = - \frac{1}{12} - \frac{\frac{49}{36}}{4 (- \frac{1}{12})}$

Step 1.2.4.2.1.2

Simplify the denominator.

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Step 1.2.4.2.1.2.1

Multiply $- 1$ by $4$ .

$e = - \frac{1}{12} - \frac{\frac{49}{36}}{- 4 (\frac{1}{12})}$

Step 1.2.4.2.1.2.2

Combine $- 4$ and $\frac{1}{12}$ .

$e = - \frac{1}{12} - \frac{\frac{49}{36}}{\frac{- 4}{12}}$

Step 1.2.4.2.1.3

Reduce the expression by cancelling the common factors.

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Step 1.2.4.2.1.3.1

Cancel the common factor of $- 4$ and $12$ .

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Step 1.2.4.2.1.3.1.1

Factor $4$ out of $- 4$ .

$e = - \frac{1}{12} - \frac{\frac{49}{36}}{\frac{4 (- 1)}{12}}$

Step 1.2.4.2.1.3.1.2

Cancel the common factors.

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Step 1.2.4.2.1.3.1.2.1

Factor $4$ out of $12$ .

$e = - \frac{1}{12} - \frac{\frac{49}{36}}{\frac{4 \cdot - 1}{4 \cdot 3}}$

Step 1.2.4.2.1.3.1.2.2

Cancel the common factor.

$e = - \frac{1}{12} - \frac{\frac{49}{36}}{\frac{4 \cdot - 1}{4 \cdot 3}}$

Step 1.2.4.2.1.3.1.2.3

Rewrite the expression.

$e = - \frac{1}{12} - \frac{\frac{49}{36}}{\frac{- 1}{3}}$

Step 1.2.4.2.1.3.2

Move the negative in front of the fraction.

$e = - \frac{1}{12} - \frac{\frac{49}{36}}{- \frac{1}{3}}$

Step 1.2.4.2.1.4

Multiply the numerator by the reciprocal of the denominator.

$e = - \frac{1}{12} - (\frac{49}{36} (- 1 \cdot 3))$

Step 1.2.4.2.1.5

Cancel the common factor of $3$ .

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Step 1.2.4.2.1.5.1

Factor $3$ out of $36$ .

$e = - \frac{1}{12} - (\frac{49}{3 (12)} (- 1 \cdot 3))$

Step 1.2.4.2.1.5.2

Factor $3$ out of $- 1 \cdot 3$ .

$e = - \frac{1}{12} - (\frac{49}{3 (12)} (3 \cdot - 1))$

Step 1.2.4.2.1.5.3

Cancel the common factor.

$e = - \frac{1}{12} - (\frac{49}{3 \cdot 12} (3 \cdot - 1))$

Step 1.2.4.2.1.5.4

Rewrite the expression.

$e = - \frac{1}{12} - (\frac{49}{12} \cdot - 1)$

Step 1.2.4.2.1.6

Combine $\frac{49}{12}$ and $- 1$ .

$e = - \frac{1}{12} - \frac{49 \cdot - 1}{12}$

Step 1.2.4.2.1.7

Multiply $49$ by $- 1$ .

$e = - \frac{1}{12} - \frac{- 49}{12}$

Step 1.2.4.2.1.8

Move the negative in front of the fraction.

$e = - \frac{1}{12} - - \frac{49}{12}$

Step 1.2.4.2.1.9

Multiply $- - \frac{49}{12}$ .

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Step 1.2.4.2.1.9.1

Multiply $- 1$ by $- 1$ .

$e = - \frac{1}{12} + 1 (\frac{49}{12})$

Step 1.2.4.2.1.9.2

Multiply $\frac{49}{12}$ by $1$ .

$e = - \frac{1}{12} + \frac{49}{12}$

Step 1.2.4.2.2

Combine the numerators over the common denominator.

$e = \frac{- 1 + 49}{12}$

Step 1.2.4.2.3

Add $- 1$ and $49$ .

$e = \frac{48}{12}$

Step 1.2.4.2.4

Divide $48$ by $12$ .

$e = 4$

Step 1.2.5

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $- \frac{1}{12} {(y - 7)}^{2} + 4$ .

$- \frac{1}{12} {(y - 7)}^{2} + 4$

Step 1.3

Set $x$ equal to the new right side.

$x = - \frac{1}{12} \cdot {(y - 7)}^{2} + 4$

Step 2

Use the vertex form, $x = a {(y - k)}^{2} + h$ , to determine the values of $a$ , $h$ , and $k$ .

$a = - \frac{1}{12}$

$h = 4$

$k = 7$

Step 3

Find the vertex $(h, k)$ .

$(4, 7)$

Step 4

Find $p$ , the distance from the vertex to the focus.

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Step 4.1

Find the distance from the vertex to a focus of the parabola by using the following formula.

$\frac{1}{4 a}$

Step 4.2

Substitute the value of $a$ into the formula.

$\frac{1}{4 \cdot (- \frac{1}{12})}$

Step 4.3

Simplify.

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Step 4.3.1

Cancel the common factor of $1$ and $- 1$ .

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Step 4.3.1.1

Rewrite $1$ as $- 1 (- 1)$ .

$\frac{- 1 (- 1)}{4 \cdot (- \frac{1}{12})}$

Step 4.3.1.2

Move the negative in front of the fraction.

$- \frac{1}{4 (\frac{1}{12})}$

Step 4.3.2

Combine $4$ and $\frac{1}{12}$ .

$- \frac{1}{\frac{4}{12}}$

Step 4.3.3

Cancel the common factor of $4$ and $12$ .

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Step 4.3.3.1

Factor $4$ out of $4$ .

$- \frac{1}{\frac{4 (1)}{12}}$

Step 4.3.3.2

Cancel the common factors.

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Step 4.3.3.2.1

Factor $4$ out of $12$ .

$- \frac{1}{\frac{4 \cdot 1}{4 \cdot 3}}$

Step 4.3.3.2.2

Cancel the common factor.

$- \frac{1}{\frac{4 \cdot 1}{4 \cdot 3}}$

Step 4.3.3.2.3

Rewrite the expression.

$- \frac{1}{\frac{1}{3}}$

Step 4.3.4

Multiply the numerator by the reciprocal of the denominator.

$- (1 \cdot 3)$

Step 4.3.5

Multiply $- (1 \cdot 3)$ .

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Step 4.3.5.1

Multiply $3$ by $1$ .

$- 1 \cdot 3$

Step 4.3.5.2

Multiply $- 1$ by $3$ .

$- 3$

Step 5

Find the focus.

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Step 5.1

The focus of a parabola can be found by adding $p$ to the x-coordinate $h$ if the parabola opens left or right.

$(h + p, k)$

Step 5.2

Substitute the known values of $h$ , $p$ , and $k$ into the formula and simplify.

$(1, 7)$

Step 6